1. definition: for any two integers a and b(b≠0), if the remainder of a divided by b is equal to the non-zero integer c, then the module C of A to b is equal to the remainder, and it is recorded as a≡b(modc).

2. The remainder theorem method: If (a, b)= 1, there are unique integers x and y, so that ax+by= 1. This conclusion is called Fermat's Little Theorem. If (a, b)= 1, there are unique integers x and y, so ax+by= 1. This conclusion is called euler theorem.

3. Division: Any positive integer A can be written as the sum of a series of positive integers. If these positive integers do not contain the factor b, they are called the remainder from A to B. Division is the way to find the greatest common divisor of two positive integers.

4. Complement method: Complement method is a method to find the greatest common divisor of two positive integers.

5. Decomposition prime factor method: decompose the prime factors of two numbers into the same prime factor, then the greatest common divisor of these two numbers is the product of these same prime factors.

Application of modulus calculation in real life;

1. Cryptography: In cryptography, modular operation is the key step to realize some encryption algorithms, such as RSA algorithm and discrete logarithm problem. By using modular operation, the security of information can be ensured and unauthorized access can be prevented.

2. Programming and computer science: In computer science, modular calculation is widely used in programming and algorithm design. For example, when dealing with circular queues and linked lists, modular operations are usually used to achieve circular effects.

3. Mathematics and statistics: In mathematics and statistics, modulus calculation is used to solve various problems, such as calculating the modulus of a vector, determining the rank of a matrix, and calculating the probability distribution.

4. Physics and engineering: In physics and engineering, modulus calculation is used to solve various problems, such as calculating vectors of speed, acceleration and force. In circuit design, modular operation is also used to calculate impedance and phase difference.

5. Finance and Economics: In the data analysis of finance and economics, modulus operation is used to calculate the dispersion and volatility of data. For example, standard deviation, coefficient of variation and other indicators involve modular operation.